Optimal adaptive barrier-function super-twisting nonlinear global sliding mode scheme for trajectory tracking of parallel robots

Compared to serial robots, parallel robots have potential superiorities in rigidity, accuracy, and ability to carry heavy loads. On the other hand, the existence of complex dynamics and uncertainties makes the accurate control of parallel robots challenging. This work proposes an optimal adaptive barrier-function-based super-twisting sliding mode control scheme based on genetic algorithms and global nonlinear sliding surface for the trajectory tracking control of parallel robots with highly-complex dynamics in the presence of uncertainties and external disturbances. The globality of the proposed controller guarantees the elimination of the reaching phase and the existence of the sliding mode around the surface right from the initial instance. Moreover, the barrier-function based adaptation law removes the requirement to know the upper bounds of the external disturbances, thus making it more suitable for practical implementations. The performance and efficiency of the controller is assessed using simulation study of a Stewart manipulator and an experimental evaluation on a 5-bar parallel robot. The obtained results were further compared to that of a six-channel PID controller and an adaptive sliding mode control method. The obtained results confirmed the superior tracking performance and robustness of the proposed approach.


Introduction
Parallel manipulators, also called parallel robots or parallel kinematic machines (PKM), are defined as manipulators which endeffector movement is controlled by at least two kinematic chains connected from the end-effector to the fixed base [1][2][3][4]. Compared to serial robots, parallel robots have potential superiorities in rigidity, accuracy, and ability to carry heavy loads [5]. There are many usages for parallel robots in several fields, such as motion simulators, haptic devices, micro-mechanisms and high precision machine tools [6,7], agriculture [8], industry [9], medical research [10] sectors, and also high-speed pick-and-place (PnP) applications in industrial fields [11], such as pharmacy, auto, food industries and electronics [12]. The primary problem lies in finding a solution to their coupled, and complex nonlinear dynamic models which not only can sufficiently characterize the real robotic system but also can (1) A control technique that eliminates the reaching mode and thereby guarantee system stability from its initial state thanks to its global property (2) A realistic design that considers an adaptation law to eliminate the need to know any information about the upper bounds of external disturbances; thus, making it more suitable for practical implementations. (3) A control approach with a minimum number of optimally tuned parameters using genetic algorithms.
The rest of this paper is organized as follows. Sect. 2 states the problem description and provides some preliminaries. Sect. 3 discusses the proposed control method for the high precision control of the parallel manipulator, and derives the proof of system using the Lyapunov stability criteria. Sect. 4, provides the simulation results aiming at assessing the efficiency and advantages of the suggested method in comparison with some conventional and nonlinear control methods. Finally, some conclusions are given in Sect. 5. s.

Dynamic model of the parallel robot
In this section, the dynamics of Stewart manipulator are formulated. SM is a parallel manipulator with three translational and three rotational DOFs, which consists of six prismatic legs that are connected by two plates, as shown in Fig. 1. The schematic view of a parallel robot is illustrated in Fig. 2. One of the mentioned plates is the base platform (BP) which is fixed in space, and the second plate is named as moving platform (MP), which is moving with six DOF; the remaining six prismatic joints (legs) are linked to the BP and MP by universal or spherical joints which are called 6-UPU, 6-UPS, 6-SPU, and 6-SPS manipulator types. For example, in the 6-SPU mechanism, the prismatic legs are connected to the BP and MP by spherical and universal joints, respectively. Then, the designed controller for this manipulator can be easily implemented to other types of parallel mechanisms. As shown in Fig. 2, the inertial coordinate (IC) system B XYZ is located at the BP with its origin at the geometry center of BP, the moving frame (MF) P XYZ is fixed to the MP center of mass. The three translational DOFs in this robot are translations on direction of the X, Y, Z axes, and the three rotational DOFs are the rotations about the axis BX, BY and, BZ.
Define the displacement and orientation vector as: where (x, y, z) denote the mass origin linear motions of MP with respect to IC, and (α, β, γ) represent the rotational motions with respect to X-axis, Y-axis, and Z-axis, individually. Consider the model of the dynamic equation of 6-DOF SM as [28,56]: where q is defined previously in Eq. (1); M(q) ∈ R 6×6 is the inertia matrix; V(q,q)q ∈ R 6×1 is the Coriolis and centrifugal force vector; G (q) is the six-dimensional gravitational force vector; F s (q) ∈ R 6×1 is the static friction vector; F d ∈ R 6×6 is the dynamic friction coefficient matrix; J(q) ∈ R 6×6 is the Jacobian matrix; τ ∈ R 6×1 is the input forces vector from prismatic leg's actuator; τ d is the 6 × 1 vector of bounded input force disturbance with ‖τ d ‖ ≤ d, where d is a positive scalar; M(q), G(q) and V(q,q) can be written as where M 0 (q) in Eq. (3), V 0 (q,q) in Eq. (4), and G 0 (q) in Eq. (5) (47) in Appendix A), and ΔM(q), ΔV(q,q), and ΔG(q) are the uncertain bounded parts of M(q), V(q,q), and G(q). The nominal parts are defined in Appendix A, which appears at the end of the article. The Jacobian matrix for dynamics (2) is described as: where L → i 's are the leg's vectors on the fixed-base frame; P i 's are the upper joints positions on the moving platform frame; R α = ⎡ are the standard rotation matrices about X, Y and Z axis, respectively. Then, Eq. (2) can be represented as: where u(t) = J T (q)τ is the control input; Ψ (q,q) = V(q,q)q + G(q). Then, in the presence of the uncertainties and external disturbances, it can be considered as Ψ (q,q) = Ψ 0 (q,q) + ΔΨ(q,q), where Ψ 0 (q,q) and ΔΨ (q,q) are the nominal and unknown parts of Ψ (q,q), correspondingly. Then, the dynamical equation (7) can be derived as follows: From Eq. (8), the dynamics of the system can be simplified as below: where Π(t) = − M 0 − 1 (q){F dq +F s (q) +τ d (q,q) +ΔΨ (q,q) +ΔM(q)q} is the disturbance and lumped uncertainty as unknown part of system dynamics and in Eq. (9), Λ 0 (t) = − M 0 − 1 (q)Ψ 0 (q,q) is the bounded known nonlinear part of system dynamics. Lemma 1. [57]: Assume that there exists a continuous and positive-definite functional V(t) which fulfills a differential equation for t ≥ t 0 and V(t 0 ) ≥ 0 as: in Eq. (10), α, β > 0, 0 < η < 1. Then, the functional V(t) converges to the origin in finite time (t S ) as Eq. (11): Lemma 2. [58]: Suppose that the continuous positive-definite V(t) gratifies the differential inequality for every t ≥ t 0 and V(t 0 ) ≥ 0 as follows: in Eq. (12) α, β > 0, 0 < η < 1 are two constants. Then, the functional V(t) converges to the origin in the finite time as Eq. (13):

Main results
In the SMC design procedure, the sliding surface selection remarkably affects the system's tracking performance. The sliding manifold is designed in such a way that when it reaches the origin; as a result, the system can obtain the anticipated performance. Let: (14) where e(t) and ė(t) are the trajectory tracking errors and their derivatives; q d represents the desired trajectory; q d denotes the desired trajectory derivative.

Stability analysis and controller design
In order to achieve a sliding mode control ( Fig. 3-a) with a nonlinear sliding surface for the dynamic system (9), the sliding surface can be proposed as: where λ is the positive coefficient. Then, the global sliding manifold ( Fig. 3-b) is configured as Eq. (16) to attain a robust global nonlinear sliding surface, which eliminates the reaching mode and subsequently leads to the existence of the error states on the sliding surface right from the initial instant: where Ω is a constant row vector, and ε is the positive value.

Remark 1.
Compared with the sliding manifold ∂ (t) = Ω(ė(t) − e(0)exp( − εt)), the nonlinear global sliding manifold (Eq. (16)) forces the error dynamics to reach the surface from the initial instance. Accordingly, the robust behavior of the system in the attendance of disturbances is guaranteed.
If ∂(t) = 0, then from Eq. (16), it can be written: It can be seen that Eq. (17) is the solution of the first-order differential equation (Eq. (18)): The goal of the GSMC law is for the error trajectory e(t) to achieve the sliding surface from the starting instance and move on the sliding surface to the equilibrium point. By taking the time-derivate of Eq. (16), we have: (19) where considering ë(t) =q −q d , it is obtainable from Eq. (14) and Eq. (19) that: One can obtain the equivalent control law from Eq. (21) for the error dynamics, ∂ (t) = 0 is the necessary condition to stay on the sliding manifold ∂(t), while in order to achieve the equivalent control law, the system uncertainties and external disturbances Π(t) are not taken into account. Afterward, the equivalent control is obtained as: However, there is no guarantee for satisfactory control performance if the system external disturbances and uncertainties Π(t) are considered. Hence, to remove the effects of the unwanted perturbations, an auxiliary control law should be defined. In practical terms, there is no exact information about the upper bound of system perturbations, and therefore, the term ‖ Π(t) ‖ is not easy to be characterized. Suppose that the unknown perturbations are bounded, i.e., >‖ Π(t) ‖ , where Γ is a positive unknown constant. Besides, assume that Γ is an estimation value for Γ, that is obtained by the following adaptive law: where κ is a constant and positive value. Then, the auxiliary control law expression can be stated as where k is a constant and positive value. The first part of Eq. (23) is an adaptive control law to compensate for the bounded uncertainties. The second part in Eq. (23) is a proportional control law toward the sliding manifold that is used as a feedback term to augment the system's stability and ameliorate the transient response.
By integrating Eq. (22) from the dynamics of the sliding manifold ∂(t), one can obtain an estimation of the parameter Γ as: Substituting Eq. (24) into Eq. (23) yields the following proportional-integral control law: where The determination of the design parameters k p and k i is dependent on the case. The multiplier k and κ parameters in Eq. (25) should be designed to ensure that the system states converge to the desired value. The phrase κ ∫ ⃦ ⃦ ∂(t)Ω T ⃦ ⃦ dt is considered as an integral control input which is an approximator for disturbances in order to eliminate the steady-state error. By enhancing the values of κ, the steady-state error is decreased and the adaptation law becomes faster. In order to improve the system stability and dynamic behavior of a proportional control technique, the phrase k Furthermore, this term eliminates the disturbance estimation error. For tracking error reduction, large values of k are used; however, this can lead to an increase in overshoot. Vice versa, reducing the value of k results in a small control gain and reduces the accuracy of the trajectory tracking. The complete control law can be achieved from Eq. (21) and Eq. (23) as:  (22)). By implementation of the adaptive control approach (Eq. (26)), the system trajectories (Eq. (7)) are converged to the sliding manifold ∂ = 0 and remain on it afterwards.
Proof: By substituting the control approach (26) into Eq. (20), one can obtain: To assess the stability property of the proposed control approach, we consider the Lyapunov stability theory. Let us choose the following positive-definite equation as a Lyapunov function candidate: in Eq. (28), Γ =Γ − Γ. Taking the time-derivative from ν(t) using Eq. (22) and Eq. (27) yields: since Γ > ‖Π(t)‖ and γκ < 1 , hence Eq. (29) can be stated by:  14)) are converged to the origin asymptotically.

Remark 3.
In order to avoid the chattering phenomena, the following modification to the adaptive control approach (26) can be carried out: where the operator sat (.) acts as a saturation function with the boundary layer thickness equal to δ.  [59]. A thin boundary layer may not eliminate the chattering phenomenon, and a wide boundary layer leads to a steady-state error. A super-twisting procedure is an appropriate substitute for the saturation function for the chattering phenomena avoidance without affecting the trajectory tracking performance [60,61].

Although the well-known technique to circumvent the chattering phenomenon uses a continuous saturation function instead of a discontinuous sign function; the implementation of such boundary layer technique is abolished
The super-twisting GSMC scheme can be adjusted as ,υ a =Γsgn ( where μ 0 is a positive and constant value. [61][62][63] has been used. [64] is suggested, which consists of a dynamic adaptive control gain establishing the sliding mode in a finite time.

Remark 6. For extending the suggested control technique, the barrier function adaptive sliding mode controller is investigated for robust tracking control of dynamics of parallel robot in the presence of external disturbances. A novel adaptive control input based on barrier function is designed in this section.
The disturbance terms can be estimated using the barrier function adaptive controller more proficiently, and the closedloop system becomes more stable. Using the control law Eq. (26) with: where ε is a positive coefficient [65]. Employing the adaptation law Eq. (35), the controller gain is tuned to increase until error signals reach the neighborhood ε at time t. For times bigger than t, the adaptation gain switches to the PSD barrier function which decreases the convergence region and maintains the errors in that convergence region. For the condition 0 < t ≤ t, the controller is proposed in Theorem 1. For the other condition when t > t, the barrier function adaptive control law is considered by: The error states reach the region |∂(t)| ≤ ε in finite time. The Lyapunov candidate function is constructed by: Time-derivate of Eq. (38) is found as where substituting ∂ (t) from Eq. (20) into Eq. (39), one has: Now, using Eq. (37) in the Eq. (40), one obtains: Eq. (41) is rewritten as: in Eq. (42), because Γ psb (t) ≥ |Π(t)| and ε (ε− |∂(t)|) 2 > 0, we have:

Optimization of controller parameters
In the proposed controller, there are several constants which appropriate choice can directly affect the controller's performance. For the higher performance and accuracy, genetic algorithms are applied for tuning the controller gains and parameters. The fitness function of the genetic algorithm is described as: where w i is the weighting factor. The goal of the optimization is to minimize the fitness function Eq. (44), thereby minimizing the tracking error of the controlled system.

Simulation results
In this part, without losing generality, the optimal adaptive barrier-function-based super-twisting nonlinear global sliding mode control is applied to a parallel robot based on Stewart platform Fig. 1, which system dynamics are mentioned in Eq. (2), and the performance of the control is compared with the results of an adaptive nonlinear sliding mode control (ANSMC) and a six-channel PID control method. The geometry and static characteristics of the SM containing mass, mass moment of inertia, and other specifications are provided in Appendix B ( Table 3). The genetic algorithm optimization that is implemented with fitness function Eq. (44), illustrated in Fig. 4 which, represents the best value of fitness function reached from 0.001378 to 0.001218, and the mean value converges to the best value in twelve generations. Eventually, the resulted parameters of the proposed controller are listed in Table 1.
The measurement noise is one of the parameters that has made the difference between simulation and practical implementation. Therefore, to make the results as accurate as possible, Gaussian noise with variance equal to 0.0005, a mean value of zero, and a sampling time equal to 5 ms is considered.
The tracking performances of SM (displacement and orientation of MP), and tracking errors are illustrated in Figs. 5 and 6, respectively. It is shown that the proposed control technique leads to better tracking, higher accuracy, and faster response than the compared methods. In Fig. 7, the six linear actuator forces are illustrated, which represents the better performance of the proposed control method. On the other hand, it can be concluded that by using the proposed control method, there is slight chattering in the time histories of the control signals, and that is because of the sensor noise. Fig. 8 (left) shows the time responses of the sliding surface and adaptation gain. As it is seen, the sliding surface of the proposed method starts from zero which confirms the globality of the proposed technique. Finally, in Fig. 8 (right), the total error signals of the three controllers are compared. Note that the tracking performance of the proposed approach outperforms that of the other approaches.

Experimental results
In this section, the experimental results on a 5-bar testbed are presented. The 5-bar setup is illustrated in Fig. 9. The robot is actuated by two motors which are connected to the microcontrollers through two digital drivers. The so-called drivers collect the joint positions and send the command torques to the actuators. The maximum torque of the motors is u max = 1.274 N.m. The controllers are commanded from a personal computer (PC) and also send the outputs to the PC during the control process. In the experiment, the reference signals are defined as task space trajectory, which are applied to a 5-bar robot as: x = − rsin((2πf)t +φ 0 ) + x 1 and y = − rsin((2πf)t +φ 0 ) + y 1 where r = 50.00 mm, f = 0.50, φ 0 = π 3 rad, x 1 = 0 and y 1 = 446.15 mm. The parameters of the proposed controller (33) are determined using the genetic algorithm approach and fitness function (44), and are illustrated in Table 2.   The experimental results on the 5-bar parallel robot are illustrated in Fig. 10 through 12. Fig. 10 provides the time histories of the end-effector position in both task and joint space. It shows that the tracking action is fulfilled under 0.4 (sec). In order to show more clearly the tracking performance, Cartesian error signals, as well as joint errors, are given in Fig. 11. Finally, Fig. 12 illustrates the control signals generated by the proposed controller to command the robot actuators. A video of the performance of the built 5-bar parallel robot is available at https://youtu.be/xPyZbsL1xFk.

Conclusions
This paper proposed an optimal adaptive barrier-function super-twisting sliding mode control scheme based on genetic algorithms and global nonlinear sliding surface for the trajectory tracking control of parallel robots in the presence of uncertainties and external disturbances. Attributes of the proposed approach are its global property, which eliminates the reaching phase thereby guaranteeing system stability from the initial state and the elimination of the requirement of availability of information about the upper bounds of external disturbances; which makes it more realistic for practical implementation. The proposed approach was assessed using both a simulation study and a practical implementation. The results were further compared to that of an ANSMC and PID controllers. The simulation results showed that the tracking error reached zero in less than 1.5 s, when using the proposed approach and 3 s for the ANSMC controller, whereas the error never converged to zero when using the PID controller. The experimental results showed that the error reached its minimum range in less than 0.5 s and the robot was able to perfectly follow the commanded trajectory. Our future research directions will focus on including a fault tolerant control component to mitigate faults such as loss of actuator effectiveness, lock-in-place and float faults. We will also investigate extending the proposed approach to the control of parallel robots with flexible links.
where m is the mass of MP; I X , I Y , and I Z are mass moment of inertia of MP about X, Y and, Z axis; g is the gravitational acceleration;